Subgroups — #27158777

Order 12416.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261833	C₂	1	1	yes	maximal, characteristic
2	#20261833	C₂	1	1	yes	maximal, characteristic
3	#20261741	C₂	1	1	yes	maximal, characteristic
4	#15690906	—	2	1	no	—
5	#15690906	—	2	1	no	—
6	C₂×#13095520	C₂²	1	1	yes	characteristic
7	C₂²×#11715897	C₄	1	1	yes	characteristic
8	#15690877	—	2	1	no	—
9	C₄⋊#11715897	—	2	1	no	—
10	#15690766	C₄	1	1	yes	characteristic
11	#13095523	—	4	1	no	—
12	#13095523	—	4	1	no	—
13	#13095520	—	4	1	no	—
14	C₂×#11715897	—	2	1	no	—
15	C₂×#11715897	—	4	1	no	—
16	C₂×#11715897	D₄	1	1	yes	characteristic
17	C₂×#11715897	D₄	1	1	yes	characteristic
18	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
19	#13095490	—	2	1	no	—
20	C₂²⋊Dic₉₇	C₈	1	1	yes	characteristic
21	C₄×Dic₉₇	—	2	1	no	—
22	C₂²⋊C₄×C₉₇	C₈	1	1	yes	characteristic
23	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₉₇	—	2	1	no	—
25	C₂×Dic₉₇	—	2	1	no	—
26	C₂×Dic₉₇	—	2	1	no	—
27	C₂×Dic₉₇	—	2	1	no	—
28	C₂×Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×Dic₉₇	M₁₆	1	1	yes	characteristic
30	#11715897	—	2	1	no	—
31	#11715897	—	2	1	no	—
32	#11715897	—	4	1	no	—
33	C₂²×C₉₇	—	2	1	no	—
34	C₂²×C₉₇	—	4	1	no	—
35	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₉₇	—	4	1	no	—
37	Dic₉₇	—	2	1	no	—
38	Dic₉₇	—	4	1	no	—
39	Dic₉₇	#99	1	1	yes	characteristic
40	Dic₉₇	#101	1	1	yes	characteristic
41	C₂×C₉₇	—	2	1	no	—
42	C₂×C₉₇	—	4	1	no	—
43	C₂×C₉₇	#324	1	1	yes	characteristic
44	#1328	—	97	1	no	maximal, Sylow
45	C₉₇	#1328	1	1	yes	characteristic, complemented, Sylow
46	#424	—	97	1	no	—
47	#349	—	97	1	no	—
48	#349	—	97	1	no	—
49	C₂×M₁₆	—	97	1	no	—
50	C₂²×C₈	—	97	1	no	—
51	#117	—	97	1	no	—
52	M₃₂	—	194	1	no	—
53	M₃₂	—	194	1	no	—
54	C₄⋊C₈	—	194	1	no	—
55	C₄⋊C₈	—	194	1	no	—
56	C₂²×C₄	—	97	1	no	—
57	M₁₆	—	388	1	no	—
58	C₂×C₈	—	97	1	no	—
59	C₂×C₈	—	97	1	no	—
60	C₂×C₈	—	388	1	no	—
61	C₂×C₈	—	194	1	no	—
62	C₄⋊C₄	—	194	1	no	—
63	C₂²⋊C₄	—	97	1	no	—
64	C₂²⋊C₄	#11715906	1	1	yes	characteristic
65	C₄²	—	194	1	no	—
66	C₁₆	—	388	1	no	—
67	C₁₆	—	388	1	no	—
68	C₂³	C₂×#11715906	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	97	1	no	—
71	C₂×C₄	—	97	1	no	—
72	C₂×C₄	—	194	1	no	—
73	C₂×C₄	—	194	1	no	—
74	C₂×C₄	—	194	1	no	—
75	C₈	—	388	1	no	—
76	C₈	—	194	1	no	—
77	C₈	—	194	1	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	4	1	no	—
80	C₂²	C₂²⋊#11715906	1	1	yes	characteristic
81	C₄	—	4	1	no	—
82	C₄	—	97	1	no	—
83	C₄	—	97	1	no	—
84	C₄	—	388	1	no	—
85	C₄	—	194	1	no	—
86	C₂	—	2	1	no	—
87	C₂	—	4	1	no	—
88	C₂	#20261821	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261821
commutator, prime radical	C₂³×C₉₇	C₂×C₈
Frattini	C₂³	C₂×#11715906
Fitting	C₂²⋊C₄×C₉₇	C₈
socle	C₂×C₉₇	#324
cosocle	C₂×#13095520	C₂²
hypercenter	C₂²⋊C₄	#11715906
hypocenter	C₉₇	#1328
LCS	G ▷ C₂³×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂²⋊C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
45. C₉₇	#1328	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20261833	2
#20261741	2
#1328	97